Banker’s gain on a long-dated bill: The face value of a bill due 5 years hence is ₹13800. At 5% per annum simple interest, compute the banker’s gain on discounting the bill.
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A₹ 690
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B₹ 600
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C₹ 590
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D₹ 625
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ENone of these
Answer
Correct Answer: ₹ 690
Explanation
Introduction / Context:Banker’s Gain (BG) is the difference between the Banker’s Discount (BD) and True Discount (TD). It measures the extra advantage to the banker arising because BD is calculated on the face value whereas the true loss to the holder is TD on present worth.
Given Data / Assumptions:
- A = ₹13800
- t = 5 years
- r = 5% = 0.05 per annum
Concept / Approach:Useful identities:
BD = A * r * t TD = A * (r * t) / (1 + r * t) BG = BD − TD = A * (r * t)^2 / (1 + r * t)Step-by-Step Solution:
r t = 0.05 * 5 = 0.25 BG = 13800 * (0.25)^2 / (1 + 0.25) BG = 13800 * 0.0625 / 1.25 = 13800 * 0.05 = ₹690Verification / Alternative check:Compute BD = 13800 * 0.25 = ₹3450; PW = A/(1 + 0.25) = 13800/1.25 = ₹11040; TD = A − PW = ₹2760; BG = 3450 − 2760 = ₹690 (consistent).
Why Other Options Are Wrong:Other values do not match BG = A*(r t)^2/(1 + r t) using r t = 0.25.
Common Pitfalls:Confusing BD with BG, or computing TD incorrectly by omitting the (1 + r t) term in the denominator.
Final Answer:₹ 690